# Search results

1. ### Practice Evaluate Trigonometric Expressions

Here is a diagram for $$45^{\circ}$$. This triangle is a right isosceles triangle, which means $$x=y$$. Use Pythagoras to determine their values...
2. ### Practice Evaluate Trigonometric Expressions

Consider the following equilateral within the unit circle: Notice that the $$x$$-axis bisects the triangle horizontally. In quadrant I, we have a 30°-60°-90° triangle. As each side of the entire equilateral triangle has a side length of 1 unit, what must the value of $$y$$ be? $$y$$ is the...
3. ### Practice Evaluate Trigonometric Expressions

I recommend knowing the following by heart: \sin(0^{\circ})=\sin\left(0\right)=0 \cos(0^{\circ})=\cos\left(0\right)=1 \sin(30^{\circ})=\sin\left(\frac{\pi}{6}\right)=\frac{1}{2} \cos(30^{\circ})=\cos\left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}...
4. ### Practice Three Angles

I have more time now, so I'll try to outline exactly how I approached this. Consider the following diagram: First I pictured the unit circle, and the line $$y=-\dfrac{1}{2}$$. The $$y$$-coordinate of any point on the unit circle represents the sine of the angle required to get to that point...
5. ### Practice Reference Angle

The reference angle is the smallest angle subtended by a radius $$0\le\theta\le\dfrac{\pi}{2}$$ and the $$x$$-axis, and the reference number is the shortest distance along the unit circle to the $$x$$-axis. Since the radius of the unit circle is by definition 1 unit, the reference number and...
6. ### Challenge Find the ratio of the shaded area to the area of the largest semi-circle

Consider the following diagram: What fraction of the area of the largest semicircle is the shaded area?
7. ### Practice Logistic Function

We are given: N(t)=\frac{P}{1+ae^{-bt}} We can see that: \lim_{t\to\infty}N(t)=P And from the graph, we see that $$P=5$$, and so we have: N(t)=\frac{5}{1+ae^{-bt}} Because the point $$(0,1)$$ is on the graph of the function, we know: N(0)=\frac{5}{1+a}=1\implies a=4 And so we now have...
8. ### Practice Half-life of Thorium-232

They should look like: Each unit on the horizontal axis represents a half-life. A unit on the vertical axis represents $$A_0$$.
9. ### Practice Bacteria Population

The setting have not changed. Testing...
10. ### Practice Find a & b

Okay, we have: 2=ae^{b} 8=ae^{4b} Hence: a=2e^{-b}=8e^{-4b} e^{-b}=4e^{-4b} e^{3b}=4 b=\frac{2}{3}\ln(2)\implies a=2^{\frac{1}{3}} Thus: y=2^{\frac{1}{3}}e^{\frac{2}{3}\ln(2)x}=2^{\frac{2x+1}{3}}
11. ### Practice Exponential Equations 1

0=2^x-5 2^x=5 x=\log_2(5)\approx2.321928094887362
12. ### Practice Region Bounded By Two Exponential Functions

Here is a diagram that should help:
13. ### Practice Values of x

If we take the natural log of both sides, we get: x<x\ln(10) x-x\ln(10)<0 x(1-\ln(10))<0 x>0 Here is a graph: $$y=10^x$$ is in green, and $$y=e^x$$ is in blue. Suppose we are given: a^x<b^x where $$a<b$$...can you algebraically solve this inequality?
14. ### Practice Moe and Larry

One way to solve this would be to graph the two lines and read off the point of intersection: Another way is to equate the two functions and solve for $$t$$: 20t+35=10t+60 10t=25 t=\frac{5}{2} We find that this problem is ill-designed in that the value we obtain for $$t$$ is not an...
15. ### Practice Functions Have An Inverse?

Let's look at a graph of the first one: What is it we're looking for to determine if the function has an inverse?
16. ### Practice Exponential Functions l

Here is a plot of all 3 functions: The one in bold red is $$y=e^x$$. The one in green is $$y=e^{-x}$$, which as I stated is a reflection of the original across the $$y$$-axis. The one in blue is $$y=-e^x$$ which is a reflection of the original across the $$x$$-axis.
17. ### Practice Exponential Function 1

To follow up: 1.) y=e^{-x} Domain: (-\infty,\infty) Range: (0,\infty) Intercept(s): (0,1) Asymptotes: y=0 2. y=-e^{-x} Domain: (-\infty,\infty) Range: (-\infty,0) Intercept(s): (0,-1) Asymptotes: y=0
18. ### Practice Exponential Function Prove 2

Consider the following rectangle: This is a "golden rectangle" because: \frac{x}{y}=\varphi where $$\varphi$$ is the golden ratio. The golden rectangle has the property such that when a square is cut off (shaded in red) the remaining rectangle is similar to the original. And so we may...
19. ### Practice Exponential Functions

A. Domain: (-\infty,\infty) Range: (-\infty,3) Intercepts: (0,2),\,(1,0) Aymptote: y=3 B. See what you can do. :)
20. ### Practice Analyze Rational Function

The only issue I see with your sketch is that the function is negative in between its roots. :) Oh, and the leftmost branch isn't shown.